Nonparametric Regression and Gaussian Random Paths

Nonparametric Regression (kernel regression)

: alphabar = (Kxz'*Kxz + 1E-4*eye(length(xu)))\Kxz'*Y;

[Change 1E-4 to different numbers.]

ccc

global click_pos flag key_pressed

click_pos = [0 0]; key_pressed = ''; flag = 'plot';

xtest = linspace(0, 10, 100)';

ytest = zeros(100, 1);

xu    = linspace(0, 10, 5)'; % Inducing points

kfun  = @(X1, X2)(kernel_se(X1, X2, struct('g2', 1, 'A', 1/(2)^2, 'w2', 0E-6, 'mzflag', 0)));

fig = figure(1);

set(fig, 'WindowButtonDownFcn', @buttonDownCallback,'KeyPressFcn', @keyDownListener ...

    , 'Position', [200 200 800 500], 'NumberTitle', 'off', 'Name', 'Kernel Regression');

ppos = zeros(1E3, 2); nrppos = 0; 

while ~isequal(flag, 'terminate') && ishandle(fig)

    % Click handler 

    switch flag 

        case 'normal' % left click

            nrppos = nrppos + 1; 

            ppos(nrppos, :) = click_pos;

            flag = 'update';

        case 'alt'

            if nrppos > 0, nrppos = nrppos -1; end;

            flag = 'update';

    end

    % Find Nonlinear Function

    if isequal(flag, 'update') 

        X        = ppos(1:nrppos, 1); 

        Y        = ppos(1:nrppos, 2); 

        Kxz      = kfun(X, xu);

        alphabar = (Kxz'*Kxz + 1E-4*eye(length(xu)))\Kxz'*Y;

        Ktest    = kfun(xtest, xu); 

        ytest    = Ktest*alphabar;

        flag = 'plot';

    end

    % Plot

    if isequal(flag, 'plot') 

        clf; hold on; 

        hp = plot(ppos(1:nrppos, 1), ppos(1:nrppos, 2), 'bo', 'MarkerSize', 15, 'LineWidth', 2);

        plot(xtest, ytest, 'k', 'LineWidth', 3);

        xlabel('x (input)', 'FontSize', 13); ylabel('f(x) (output)', 'FontSize', 13);

        axis equal; axis([0 10 -3 3]); grid on;

        title(sprintf('Kernel Regression with %d Positions', nrppos));

        drawnow;

        flag = 'waiting';

    end

    if isequal(key_pressed, 'q')

        flag = 'terminate';

    else

        key_pressed = '';

    end

    pause(1E-10);

end

fprintf(2, 'Terminated. \n');

Gaussian Random Paths (sampling a diverse set of paths given waypoints)

ccc

global click_pos flag key_pressed

click_pos = [0 0]; key_pressed = ''; flag = 'plot';

fig = figure(1);

set(fig, 'WindowButtonDownFcn', @buttonDownCallback,'KeyPressFcn', @keyDownListener ...

    , 'Position', [200 200 800 500], 'NumberTitle', 'off', 'Name', 'Gaussian Random Paths');

ppos = zeros(1E3, 2); nrppos = 0; emsec = 0; nrPath = 0;

while ~isequal(flag, 'terminate') && ishandle(fig)

    % Click handler 

    switch flag 

        case 'normal' % left click

            nrppos = nrppos + 1; 

            ppos(nrppos, :) = click_pos;

            flag = 'update';

        case 'alt'

            if nrppos > 0, nrppos = nrppos -1; end;

            flag = 'update';

    end

    % Find connecting GRP

    if isequal(flag, 'update') 

        xy = ppos(1:nrppos, :);

        if nrppos >= 2

            ts = get_ts(xy); xs = xy(:, 1); ys = xy(:, 2);

            testInputs = linspace(0, ts(end), 50)';

            kfun   = @(X1, X2)(kernel_se(X1, X2, struct('g2', 5^2, 'A', 1/(5)^2, 'w2', 0, 'mzflag', 0)));

            nrPath = 1000;

            iclk = clock;

            [xPaths, xMeanPath] = grp(ts, xs, testInputs, kfun, nrPath);

            [yPaths, yMeanPath] = grp(ts, ys, testInputs, kfun, nrPath);

            emsec = etime(clock, iclk)*1000;

        end

        flag = 'plot';

    end

    % Plot

    if isequal(flag, 'plot') 

        clf; hold on; 

        if nrppos >= 2

            colors = copper(nrPath);

            for i = 1:nrPath

                plot(xPaths(i, :), yPaths(i, :), '-', 'Color', colors(i, :));

            end

            plot(xMeanPath, yMeanPath, 'b--', 'LineWidth', 3);

        end

        if nrppos > 0, hp = plot(xy(:, 1), xy(:, 2), 'bo', 'MarkerSize', 11, 'LineWidth', 2); end

        xlabel('X', 'FontSize', 13); ylabel('Y', 'FontSize', 13);

        axis equal; axis([0 10 0 5]); grid on; title(sprintf('Sampled %d paths in %.1fms', nrPath, emsec));

        drawnow;

        flag = 'waiting';

    end

    if isequal(key_pressed, 'q')

        flag = 'terminate';

    else

        key_pressed = '';

    end

    pause(1E-10);

end

fprintf(2, 'Terminated. \n');

Matlab code

kos.zip